If - and - are the roots of the quadratic equation a x2-b x-c-0- then lim x -1-1-cosc x2-b x-a-21- x2-A- -c-2 -1-1-B- c-1-1-C- c-2 -1-1-D- -2 c1-1- -

The correct option is A | c/2 α ( 1/α 1/β ) |α and β are the roots of the equation ax^2+bx+c=0 ∴α + β = b/a and αβ = c/a Let α' and β' be the roots of the eq ...

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Standard XII

Mathematics

Common Roots

If α and β ar...

Question

If $α$ and $β$ are the roots of the quadratic equation $a x^{2} + b x + c = 0$ , then $lim x \to \frac{1}{α} \sqrt{\frac{1 - cos (c x^{2} + b x + a)}{2 (1 - α x)^{2}}} =$

∣ ∣ \frac{c}{2 α} (\frac{1}{α} - \frac{1}{β}) ∣ ∣

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∣ ∣ \frac{c}{2 β} (\frac{1}{α} - \frac{1}{β}) ∣ ∣

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∣ ∣ \frac{c}{α β} (\frac{1}{α} - \frac{1}{β}) ∣ ∣

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∣ ∣ \frac{α}{2 c} (\frac{1}{α} - \frac{1}{β}) ∣ ∣

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Solution

The correct option is A $∣ ∣ \frac{c}{2 α} (\frac{1}{α} - \frac{1}{β}) ∣ ∣$
$α$ and $β$ are the roots of the equation $a x^{2} + b x + c = 0$
$∴ α + β = - \frac{b}{a} and α β = \frac{c}{a}$
Let $α^{'}$ and $β^{'}$ be the roots of the equation $c x^{2} + b x + a = 0$
$∴ α^{'} + β^{'} = - \frac{b}{c} and α^{'} β^{'} = \frac{a}{c}$
Comparing these we get $α^{'} = \frac{1}{α} and β^{'} = \frac{1}{β}$
$∴ c x^{2} + b x + a = c (x - \frac{1}{α}) (x - \frac{1}{β})$

$∴ lim x \to \frac{1}{α} \sqrt{\frac{1 - cos (c x^{2} + b x + a)}{2 (1 - α x)^{2}}}$
$= lim x \to \frac{1}{α} \sqrt{\frac{1 - cos (c (x - \frac{1}{α}) (x - \frac{1}{β}))}{2 α^{2} (x - \frac{1}{α})^{2}}}$
$= lim x \to \frac{1}{α} \sqrt{\frac{2 {sin}^{2} (\frac{c}{2} (x - \frac{1}{α}) (x - \frac{1}{β}))}{2 α^{2} (x - \frac{1}{α})^{2}}}$
$= lim x \to \frac{1}{α} \sqrt{\frac{1}{α^{2}} [\frac{{sin}^{2} (\frac{c}{2} (x - \frac{1}{α}) (x - \frac{1}{β}))}{{\frac{c}{2} (x - \frac{1}{α}) (x - \frac{1}{β})}^{2}}] \cdot (x - \frac{1}{β})^{2} \cdot \frac{c^{2}}{4}}$
$= lim x \to \frac{1}{α} \sqrt{\frac{1}{α^{2}} \cdot \frac{c^{2}}{4} \cdot (x - \frac{1}{β})^{2}}$
$= ∣ ∣ \frac{c}{2 α} (\frac{1}{α} - \frac{1}{β}) ∣ ∣$

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